How do you show a number that large number is prime?

A prime is an integer that has only one and itself for positive divisors (for
example, the prime factors of 10 are 2 and 5). The first few primes are 2, 3, 5, 7 and 11.
We can check small numbers by just dividing by the lesser integers. But
to do that with a number this size would take far longer than the universe has
been in existence. The key to slaying a giant this size is a theorem developed
by Lucas
in the late 1870's and simplified by Lehmer, now called the Lucas-Lehmer Test.
Even as simple as this test is, it could not be done quickly without using very
clever programs to multiply the numbers. In 1968 Strassen discovered how to multiply
quickly using
Fast Fourier
Transforms (*). He and Schönhage
refined and published the method in 1971. GIMPS now uses an improved version of
their algorithm developed by the long time Mersenne searcher Richard
Crandall
[see CF94].

Is that it, just get a fast program?

Even a fast program is not enough, alone a systematic search would take nearly
a millennium on Spence's computer. What GIMP's does is to coordinate the
efforts of over two thousand computer users--by
offering free software, a database of known results, and by
allowing individuals to check out regions to test. Gordon Spence
and George Woltman share the credit and glory with all of the others
involved in this effort! There are still infinitely many more giants left
to slay, so why not surf over to Woltman's
GIMPS site and join the search for the next record prime?